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Lehmer's number is not an orientable stretch factor in genus six

  • Lizi Guo
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Abstract

Let λL ≈ 1.17628 be Lehmer's number. We prove that no pseudo-Anosov homeomorphism of a closed orientable surface of genus six with orientable invariant foliations has stretch factor λL. This gives a negative answer to a question of Lanneau and Thiffeault on the minimal stretch factor in even genus. As a consequence, the minimal stretch factor δ6+ of such maps is the largest root of x12−x9−x7+x6−x5−x3+1, approximately $1.20189$. The proof combines the Lefschetz method of Lanneau and Thiffeault, the standardly embedded train tracks of Hironaka and Tsang, an elementary analysis of the transition digraph, and an obstruction from periodic conjugacy classes in the fundamental group.

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Cite

BibTeXRIS
BibTeX
@misc{guo2026lehmers,
  title = {{Lehmer's number is not an orientable stretch factor in genus six}},
  author = {Lizi Guo},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01350},
  eprint = {2611.01350},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01350},
}
RIS
TY  - UNPB
TI  - Lehmer's number is not an orientable stretch factor in genus six
AU  - Lizi Guo
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01350
UR  - https://arxiv.org/abs/2611.01350
AB  - Let λ_L ≈ 1.17628 be Lehmer's number. We prove that no pseudo-Anosov homeomorphism of a closed orientable surface of genus six with orientable invariant foliations has stretch factor λ_L. This gives a negative answer to a question of Lanneau and Thiffeault on the minimal stretch factor in even genus. As a consequence, the minimal stretch factor δ_6^+ of such maps is the largest root of x^12−x^9−x^7+x^6−x^5−x^3+1, approximately $1.20189$. The proof combines the Lefschetz method of Lanneau and Thiffeault, the standardly embedded train tracks of Hironaka and Tsang, an elementary analysis of the transition digraph, and an obstruction from periodic conjugacy classes in the fundamental group.
ER  -